{"title":"Robust DG Schemes on Unstructured Triangular Meshes: Oscillation Elimination and Bound Preservation via Optimal Convex Decomposition","authors":"Shengrong Ding, Shumo Cui, Kailiang Wu","doi":"arxiv-2409.09620","DOIUrl":null,"url":null,"abstract":"Discontinuous Galerkin (DG) schemes on unstructured meshes offer the\nadvantages of compactness and the ability to handle complex computational\ndomains. However, their robustness and reliability in solving hyperbolic\nconservation laws depend on two critical abilities: suppressing spurious\noscillations and preserving intrinsic bounds or constraints. This paper\nintroduces two significant advancements in enhancing the robustness and\nefficiency of DG methods on unstructured meshes for general hyperbolic\nconservation laws, while maintaining their accuracy and compactness. First, we\ninvestigate the oscillation-eliminating (OE) DG methods on unstructured meshes.\nThese methods not only maintain key features such as conservation, scale\ninvariance, and evolution invariance but also achieve rotation invariance\nthrough a novel rotation-invariant OE (RIOE) procedure. Second, we propose, for\nthe first time, the optimal convex decomposition for designing efficient\nbound-preserving (BP) DG schemes on unstructured meshes. Finding the optimal\nconvex decomposition that maximizes the BP CFL number is an important yet\nchallenging problem.While this challenge was addressed for rectangular meshes,\nit remains an open problem for triangular meshes. This paper successfully\nconstructs the optimal convex decomposition for the widely used $P^1$ and $P^2$\nspaces on triangular cells, significantly improving the efficiency of BP DG\nmethods.The maximum BP CFL numbers are increased by 100%--200% for $P^1$ and\n280.38%--350% for $P^2$, compared to classic decomposition. Furthermore, our\nRIOE procedure and optimal decomposition technique can be integrated into\nexisting DG codes with little and localized modifications. These techniques\nrequire only edge-neighboring cell information, thereby retaining the\ncompactness and high parallel efficiency of original DG methods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09620","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Discontinuous Galerkin (DG) schemes on unstructured meshes offer the
advantages of compactness and the ability to handle complex computational
domains. However, their robustness and reliability in solving hyperbolic
conservation laws depend on two critical abilities: suppressing spurious
oscillations and preserving intrinsic bounds or constraints. This paper
introduces two significant advancements in enhancing the robustness and
efficiency of DG methods on unstructured meshes for general hyperbolic
conservation laws, while maintaining their accuracy and compactness. First, we
investigate the oscillation-eliminating (OE) DG methods on unstructured meshes.
These methods not only maintain key features such as conservation, scale
invariance, and evolution invariance but also achieve rotation invariance
through a novel rotation-invariant OE (RIOE) procedure. Second, we propose, for
the first time, the optimal convex decomposition for designing efficient
bound-preserving (BP) DG schemes on unstructured meshes. Finding the optimal
convex decomposition that maximizes the BP CFL number is an important yet
challenging problem.While this challenge was addressed for rectangular meshes,
it remains an open problem for triangular meshes. This paper successfully
constructs the optimal convex decomposition for the widely used $P^1$ and $P^2$
spaces on triangular cells, significantly improving the efficiency of BP DG
methods.The maximum BP CFL numbers are increased by 100%--200% for $P^1$ and
280.38%--350% for $P^2$, compared to classic decomposition. Furthermore, our
RIOE procedure and optimal decomposition technique can be integrated into
existing DG codes with little and localized modifications. These techniques
require only edge-neighboring cell information, thereby retaining the
compactness and high parallel efficiency of original DG methods.